How can motion ever start? I've recently been puzzled by a question that I hope you can help me figure out. I heard about Zeno's Paradoxes a while back, and I recently asked myself something which is similar but not quite the same. 
Assume we have a car, or some object starting at rest with respect to some coordinate system. In order for this object to move its position, it has to gain first some velocity; however, this in turn requires some acceleration. Now, I know that force is all that is needed to accelerate some object because $F = ma$, but an applied force doesn't go from $0$ to something nonzero instantly, right? So there must be some other $dF/dt$ that depends on some other variable $G$, and that $G$ would also have some dependency $dG/dt$ on some other variable and so on:
\begin{align}
dx/dt &= v \\
dv/dt &= F/m \\
dF/dt &= G \\
dG/dt &= H \\
\end{align}


*

*how can there be motion if this system of ODEs can be made arbitrarily long?

*Since I am able to move objects, does that mean that the system of ODEs caps at some higher order derivative? If so, which one?
 A: Your conclusion is partly right. It's true that if a force is analytic (i.e. equal to its Taylor series) and all of its derivatives are zero at some point, then it must be zero everywhere. Then an object can never get started moving.
However, forces in classical mechanics are rarely like this. For example, the gravitational force is proportional to $1/r^2$, so every single term of your expansion already has nonzero terms, at all times. We can also consider contact forces, like the force between two billiard balls as they collide. Realistically, such a force should be modeled by considering electromagnetic interactions, which are spread out just like gravity. So the real force may be something like $1/r^6$, which is still nonzero at all points.
We can also consider nonanalytic forces. For example, we might say that a contact force suddenly turns on once objects get closer than $r_0$ apart. However, in this case $F$ is not differentiable, so your tower of differential equations doesn't exist.
A: Mathematically speaking let your position be a function of time and lets say it is given by $t^3$ now as your question says ,this will contribute up to third order derivative but each time the magnitude decreases e.g. $t^2$,$t^1$,$t^0$...so we do not need to consider them.
Moreover the coefficient associated with higher order terms of position in nature is low,  meaning the higher order terms generally do not appear this is also the reason in generalised study of classical mechanics we take F=F($x$,$dx/dt$, $t$) even neglecting $d^2x/dt^2$ term.
